Financial Mathematics Text

Thursday, February 13, 2014

Return on Invested Time: Developing Understanding

Time is our most valuable possession; and, yet, we trade it for other things.
One thing I noticed relatively early on is my ability to process information better than others. I don't know to what extent this is genetic or learned but I do think part of it comes down to how I engage information.

Many tasks in my early education experience I considered to be pointless or a waste of time. While I jokingly referred to myself as a slacker, to some extent it was deliberate in that I saw some activities as being unproductive and would rather spend my time doing activities either more enjoyable or more productive.

In hindsight, I think I may have misjudged in some cases but in many cases my approach has been successful. Here's a look at my overall approach.

I wish I could Remember

Tomorrow is the big test and I need to retain as much information as possible. I stare at my notes and book and commit to memory as much as I can. This is done to increase my test score. Hopefully I can retain enough information to pass this test.

It is said that philosopher and theologian Thomas Aquinas committed to memory the entire Bible and several of Aristotle's books. If only I could have such memory!

And there are a number of ways to improve memory. There's been a considerable amount of research on memory in cognitive psychology. Some of this research can lead you to improve your memory.

So I employ some of these techniques. Come test time, I am able to recall all sorts of information for the test!

By the end of the day, I have forgotten some of it. After a week I've forgotten a bit more. As months and even years go by, the amount of information I have retained continues to diminish.

Memorization: A Good Return on Investment?

At the time Thomas Aquinas lived, books were primarily transcribed by monks. This resulted in very limited access. Spending time memorizing may very well have been a fruitful activity for him.

In 1439, Gutenberg is credited with inventing the printing press. This invention revolutionized the Western world. One could very well argue that this invention was partly responsible for things like the Protestant reformation and the "scientific revolution".

Having books readily available significantly reduces the value of having entire books committed to memory. I need not memorize a passage but rather look up the information.

In the internet age, we have a wealth of information at our fingertips. With a good internet connection, I can quickly look up any bit of information rather quickly. What need do I have to memorize it?

The key, I think, is knowing how to find information that I need. So I need to know where to look in the book (or how to use the table of contents or index) or what search terms to plug into Google. 

What's worth "memorizing"?

So what's actually worth remembering? The short answer: not much.

As I alluded to, remembering how to look up information is useful. But I don't think that's a purely memorization task. It's knowing a technique. Sometimes finding what I want, say by a Google search, requires a few trial and error guesses on what to look for. It takes a certain skill set to be able to do this well.

But I also think there's a broad, what I would refer to, as an axiomatic approach. . .

The Axiomatic Approach

What I call the Axiomatic Approach amounts to memorizing a few key principles ("axioms") and knowing how to deduce from them (in, perhaps, a less than formal sense) other facts. It's a way of seeing how ideas sort of hang together.

Now it doesn't have to be done in a foundationalist sense; it could be more along the lines of a coherent whole. But the key is focusing on those few ideas and techniques which have a broad range of applications. By doing so you get a good return on investment.

Example: The Quadratic Equation

For some reason, many teachers think this formula needs to be memorized. Sure, if you use often, it can be convenient to memorize. But if you use it often, you'll probably memorize it without much effort.

In practice, you should be able to look the formula up quickly if you need to do so, so why memorize it? Or is there a better alternative?

Here's a far more useful exercise. But we'll need a tool to do this.

Completing the Square

Completing the square is a technique that begins with an expression of the form $ax^2 + bx + c$ and puts it in the form  $a(x+h)^2 + k$. With this technique we can actually derive the quadratic equation.

Deriving the Quadratic Equation

Ultimately we have something of the form $ax^2 + bx + c = 0$ and our goal is to solve for $x$. Assuming $a\neq0$ we can divide by sides by $a$:
$$x^2 +\frac{b}{a}x + \frac{c}{a} = 0$$
Next we "complete the square":
$$(x+\frac{b}{2a})^2 + \frac{c}{a} - \frac{b^2}{4a^2} = 0$$
Next we'll combine some fractions and move them to the right side:
$$ (x+\frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2} $$
We can take the square root of both sides:
$$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$
And then solving for $x$ we arrive at the quadratic equation.
$$x = \frac{-b  \pm \sqrt{b^2 - 4ac}}{2a}$$
As it turns out, you can actually use the completing the square technique to simply solve any quadratic equation without explicitly using the formula. 

Concluding Thoughts

Developing understanding takes a slightly different skill set than memorization. But I believe it to be a far more useful way of spending one's time. Understanding allows one to derive a large number of facts and principles and can be applied to a wide range of areas.

Memorization, on the other hand, often doesn't last very long and can be easily replaced by a good book or your favorite search engine. If you have some time to kill, maybe memorization might provide for some amusement. For the rest of us, understanding is far more rewarding.

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